A bijection between fractional trees and d -angulations Marie - - PowerPoint PPT Presentation

A bijection between fractional trees and d-angulations

Marie Albenque and Dominique Poulalhon

LIX – CNRS

Young workshop in arithmetics and combinatorics – June, 22th 2011

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 1 / 14

Definition of planar maps

Planar map = planar connected graph embedded properly in the sphere up to a direct homomorphism of the sphere Rooted planar map = an oriented edge is marked. with a planar embedding = the “outer face” is chosen.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 2 / 14

Triangulations, quadrangualations, . . .

Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d-angulation, . . . = map whose faces are all of degree 3, 4, 5, d, . . . Girth = length of the shortest cycle. From now on, only d-angulations of girth d.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 3 / 14

Triangulations, quadrangualations, . . .

Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d-angulation, . . . = map whose faces are all of degree 3, 4, 5, d, . . . Girth = length of the shortest cycle. From now on, only d-angulations of girth d.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 3 / 14

Triangulations, quadrangualations, . . .

Faces = connected components of the plane without the edges of the map. Triangulation, quadrangulation, pentagulation, d-angulation, . . . = map whose faces are all of degree 3, 4, 5, d, . . . Girth = length of the shortest cycle. From now on, only d-angulations of girth d.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 3 / 14

Enumeration

One of the main question when studying some families of maps : How many maps belong to this family ? Tutte ’60s: recursive decomposition Matrix integrals: t’Hooft ’74,Brézin, Itzykson, Parisi and Zuber ’78 , Representation of the symmetric group: Goulden and Jackson ’87 , Bijective approach with labeled trees: Cori-Vauquelin ’81, Schaeffer ’98, Bouttier, Di Francesco and Guitter ’04, Bernardi, Chapuy, Fusy, Miermont, . . . Bijective approach with blossoming trees: Schaeffer ’98, Schaeffer and Bousquet-Mélou ’00, Poulalhon and Schaeffer ’05, Fusy, Poulalhon and Schaeffer ’06.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 4 / 14

Rooted simple triangulations

The number of rooted simple triangulations with 2n faces, 3n edges and n + 2 vertices is equal to: 2(4n − 3)! n!(3n − 1)! = 1 n · 2 (4n − 2) 4n − 2 n − 1

• number of blossoming trees

with n nodes

. Blossoming tree = rooted plane tree where each node (= inner vertex) carries exactly two leaves.

Theorem (Poulalhon and Schaeffer ’05)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 5 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

Root of the tree is not involved in the local closure ⇒ the tree is balanced.

n trees correspond to the same rooted triangulation.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Closure of a blossoming tree

How to describe the inverse construction ? with orientations.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 6 / 14

Orientations

Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α-orientation (Felsner). For triangulations: 3-orientation =

• ut(v) = 3

for each v not in the root face

• ut(v) = 0
• therwise.

Theorem (Schnyder ’89, Felsner ’04)

Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the

• rientation is accessible.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 7 / 14

Orientations

Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α-orientation (Felsner). For triangulations: 3-orientation =

• ut(v) = 3

for each v not in the root face

• ut(v) = 0
• therwise.

Theorem (Schnyder ’89, Felsner ’04)

Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the

• rientation is accessible.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 7 / 14

Orientations

Orientation of a planar map = an orientation is given to each edge We want to consider orientations where the outdegree of each vertex is prescribed → general theory of α-orientation (Felsner). For triangulations: 3-orientation =

• ut(v) = 3

for each v not in the root face

• ut(v) = 0
• therwise.

Theorem (Schnyder ’89, Felsner ’04)

Each rooted triangulation of girth 3 admits a unique minimal 3-orientation, ie. a 3-orientation without counterclockwise cycle. Moreover there exists a directed path from any vertices to the root face : the

• rientation is accessible.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 7 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

Inverse construction

Theorem (Poulalhon and Schaeffer ’98)

There exists a one-to-one correspondence between the set of balanced plane trees with n nodes and two leaves adjacent to each node, and the set of rooted simple triangulations of size n.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 8 / 14

And for d-angulations ?

k-fractional orientation = orientation of the expended map where each edge is replaced by k copies. j/k-orientation =

• ut(v) = j

for each v not in the root face

• ut(v) = k
• therwise.

Theorem (Bernardi and Fusy ’11)

Any rooted d-angulation of girth d admits a unique minimal

d d−2-orientation such

that the root face is a clockwise cycle. Moreover this orientation is accessible.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 9 / 14

d-fractional trees

d-fractional tree = rooted plane tree where each edge carries a flow (possibly in two directions) such that: sum of the flows in the edge = d − 2, for each node u, out(u) = d, for each leaf l, out(l) = 0, there exists a directed path from each node to the root. → Trees not stable by rerooting, do not lead to nice combinatorial equalities. ⇒ Cyclic closure operation d-fractional forest = simple rooted cycle of length d, on which are grafted d-fractional trees.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 10 / 14

d-fractional trees

d-fractional tree = rooted plane tree where each edge carries a flow (possibly in two directions) such that: sum of the flows in the edge = d − 2, for each node u, out(u) = d, for each leaf l, out(l) = 0, there exists a directed path from each node to the root. → Trees not stable by rerooting, do not lead to nice combinatorial equalities. ⇒ Cyclic closure operation d-fractional forest = simple rooted cycle of length d, on which are grafted d-fractional trees.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 10 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Closure of a d-fractional forest

1 2 2 2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 2 2 2 2

Theorem

There exists a one-to-one constructive correspondence between d-fractional forests with n nodes and rooted d-angulations of girth d with n vertices.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 11 / 14

Proof of the theorem

Induction on the number of faces of M. There exists a saturated clockwise edge e on the outer face:

1

if M\e is still accessible: delete e.

2

• therwise, there exists such a partition:

e

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 12 / 14

Proof of the theorem

Induction on the number of faces of M. There exists a saturated clockwise edge e on the outer face:

1

if M\e is still accessible: delete e.

2

• therwise, there exists such a partition:

e lmin f

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 12 / 14

Proof of the theorem

Induction on the number of faces of M. There exists a saturated clockwise edge e on the outer face:

1

if M\e is still accessible: delete e.

2

• therwise, there exists such a partition:

e l < lmin f

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 12 / 14

Generalization

“Theoretical proof” in quadratic time: relying on it, we can give a direct method to identify the closure edges. ⇒ Opening algorithm in linear time. Method generalizes directly to p-gonal d-angulations (ie. map with faces of degree d but root face of degree p). Enumerative consequences: recursive decomposition of the d-fractional trees ⇒ Equations for the generating series of d-angulations. General framework to obtain a bijection between maps endowed with a minimal accessible orientation and blossoming trees. ⇒ Yield enumerative results when the blossoming trees can be enumerated.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 13 / 14

Generalization

“Theoretical proof” in quadratic time: relying on it, we can give a direct method to identify the closure edges. ⇒ Opening algorithm in linear time. Method generalizes directly to p-gonal d-angulations (ie. map with faces of degree d but root face of degree p). Enumerative consequences: recursive decomposition of the d-fractional trees ⇒ Equations for the generating series of d-angulations. General framework to obtain a bijection between maps endowed with a minimal accessible orientation and blossoming trees. ⇒ Yield enumerative results when the blossoming trees can be enumerated.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 13 / 14

Generalization

“Theoretical proof” in quadratic time: relying on it, we can give a direct method to identify the closure edges. ⇒ Opening algorithm in linear time. Method generalizes directly to p-gonal d-angulations (ie. map with faces of degree d but root face of degree p). Enumerative consequences: recursive decomposition of the d-fractional trees ⇒ Equations for the generating series of d-angulations. General framework to obtain a bijection between maps endowed with a minimal accessible orientation and blossoming trees. ⇒ Yield enumerative results when the blossoming trees can be enumerated.

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 13 / 14

That’s all . . . Thank you !

Albenque & Poulalhon (LIX – CNRS) Bijection for d-angulations Madrid, June 22th 2011 14 / 14